39.3COMar 17
The independence ratio of 4-cycle-free planar graphsTom Kelly, Sid Kolichala, Caleb McFarland et al.
We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/α(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to $\varepsilon = 2/9$.